# Random Structures & Algorithms

Random Structures & Algorithms > 55 > 2 > 371 - 401

Random Structures & Algorithms > 55 > 2 > 249 - 270

**321**‐avoiding permutations of length

*n*and show that the number of occurrences of another pattern

*σ*has a limit distribution, after scaling by

*n*

^{m + ℓ}where

*m*is the length of

*σ*and

*ℓ*is the number of blocks in it. The limit is not normal, and can be expressed as a functional of a Brownian excursion.

Random Structures & Algorithms > 55 > 2 > 427 - 495

Random Structures & Algorithms > 55 > 2 > 290 - 312

*G*obtained by starting with a single vertex of

*G*and repeatedly selecting, uniformly at random, an edge of

*G*with exactly one endpoint in

*T*and adding this edge to

*T*.

Random Structures & Algorithms > 55 > 2 > 342 - 355

*S*

_{n}and are consistent as

*n*varies. The extreme infinitely spherically symmetric permutation‐valued processes are identified for the Hamming, Kendall‐tau and Cayley metrics. The proofs in all three cases are based on a unified approach through stochastic monotonicity.

Random Structures & Algorithms > 55 > 2 > 356 - 370

*X*

_{1},

*X*

_{2},… with finite mean

*μ*and standard deviation

*σ*such that

*μ*is the solution for the problem input, and the relative standard deviation |

*σ*/

*μ*| ≤

*c*for known

*c*....

Random Structures & Algorithms > 55 > 2 > 496 - 528

*n*vertices and vertex degrees in a fixed set converges in the Gromov‐Hausdorff sense after a suitable rescaling to the Brownian continuum random tree. This confirms a conjecture by Aldous (1991). We also establish Benjamini‐Schramm convergence of this model of random trees and provide a general approximation result, that allows for a transfer of...

Random Structures & Algorithms > 55 > 2 > 313 - 341

Random Structures & Algorithms > 55 > 2 > 271 - 289

*μn*‐bounded edge colorings of Dirac bipartite graphs, for a sufficiently small

*μ*> 0. As an application of our results, we obtain several results on the existence of rainbow

*k*‐factors in Dirac graphs and rainbow spanning subgraphs of bounded maximum degree on graphs with large minimum degree.

Random Structures & Algorithms > 55 > 2 > 402 - 426

*k*], iff there are indices

*i*

_{1}< … <

*i*

_{k}, such that

*f*(

*i*

_{x}) >

*f*(

*i*

_{y}) whenever

*π*(

*x*) >

*π*(

*y*). Otherwise,

*f*is

*π*‐free. We study the property testing problem of distinguishing, for a fixed

*π*, between

*π*‐free sequences and the sequences which differ from any

*π*‐free sequence in more than

*ϵ n*places. Our main findings are as follows:...

Random Structures & Algorithms > 55 > 3 > 584 - 614

*n*‐vertex

*d*‐dimensional cube of the integer lattice graph ${Z}^{d}$. We study the effects that the strong spatial mixing condition (SSM) has on the rate of convergence to equilibrium of

*nonlocal*Markov chains. We prove that when SSM holds, the relaxation time (i.e., the inverse spectral gap) of general

*block dynamics*is

*O*(

*r*), where

*r*is the...

Random Structures & Algorithms > 55 > 3 > 696 - 718

*m*edges is added, exhibit any symmetry...

Random Structures & Algorithms > 55 > 3 > 560 - 583

*r*get connected by an edge with probability proportional to

*r*

^{−s}, for

*s*∈ (

*d*,2

*d*), and study the asymptotic of the graph‐theoretical (a.k.a. chemical) distance

*D*(

*x*,

*y*) between

*x*and

*y*in the limit as |

*x*−

*y*|→

*∞*. For the model on ${Z}^{d}$ we show that, in probability as |

*x*|→

*∞*, the distance

*D*(0,

*x*) is squeezed between two positive...

Random Structures & Algorithms > 55 > 3 > 677 - 695

*d*‐dimensional analog of Cayley's formula for the number of

*n*‐vertex trees. He enumerated

*d*‐dimensional hypertrees weighted by the squared size of their (

*d*− 1)‐dimensional homology group. This, however, does not answer the more basic problem of unweighted enumeration of

*d*‐hypertrees, which is our concern here. Our main result, Theorem 1.4, significantly...

Random Structures & Algorithms > 55 > 3 > 649 - 676

Random Structures & Algorithms > 55 > 3 > 742 - 771

*α*‐stable...

Random Structures & Algorithms > 55 > 3 > 719 - 741

*G*is called the cop number of

*G*. The biggest open conjecture in this area is the one of Meyniel, which asserts that for some absolute constant

*C*, the cop number of every connected graph

*G*is at most $C\sqrt{\left|V\right(G\left)\right|}$. In a separate paper, we showed...

Random Structures & Algorithms > 55 > 3 > 545 - 559

*λ*

_{1}(

*λ*

_{2}) times the number of nearest neighbors of type 1 (2). Assuming (essentially) that the degree...